Problem: Simplify the following expression and state the conditions under which the simplification is valid. You can assume that $x \neq 0$. $t = \dfrac{x^2 - 10x}{x^2 - 20x + 100} \times \dfrac{x - 10}{9x - 81} $
First factor the quadratic. $t = \dfrac{x^2 - 10x}{(x - 10)(x - 10)} \times \dfrac{x - 10}{9x - 81} $ Then factor out any other terms. $t = \dfrac{x(x - 10)}{(x - 10)(x - 10)} \times \dfrac{x - 10}{9(x - 9)} $ Then multiply the two numerators and multiply the two denominators. $t = \dfrac{ x(x - 10) \times (x - 10) } { (x - 10)(x - 10) \times 9(x - 9) } $ $t = \dfrac{ x(x - 10)(x - 10)}{ 9(x - 10)(x - 10)(x - 9)} $ Notice that $(x - 10)$ appears twice in both the numerator and denominator so we can cancel them. $t = \dfrac{ x(x - 10)\cancel{(x - 10)}}{ 9\cancel{(x - 10)}(x - 10)(x - 9)} $ We are dividing by $x - 10$ , so $x - 10 \neq 0$ Therefore, $x \neq 10$ $t = \dfrac{ x\cancel{(x - 10)}\cancel{(x - 10)}}{ 9\cancel{(x - 10)}\cancel{(x - 10)}(x - 9)} $ We are dividing by $x - 10$ , so $x - 10 \neq 0$ Therefore, $x \neq 10$ $t = \dfrac{x}{9(x - 9)} ; \space x \neq 10 $